Question

(x^2 - 6x)^2 - 2(x-3)^2=81(Решите уравнение)

Answer 1

$(x^{2} - 6x)^{2} - 2(x - 3)^{2} = 81$

$(x^{2} - 6x + 9 - 9)^{2} - 2(x - 3)^{2} - 81 = 0$

$((x - 3)^{2} - 9)^{2} - 2(x - 3)^{2} - 81 = 0$

$(x - 3)^{4} - 18(x - 3)^{2} + 81 - 2(x - 3)^{2} - 81 = 0$

$(x - 3)^{4} - 20(x - 3)^{2} = 0$

$(x - 3)^{2}((x - 3)^{2} - 20) = 0$

$(x - 3)^{2}(x^{2} - 6x + 9 - 20) = 0$

$(x - 3)^{2}(x^{2} - 6x - 11) = 0$

$1) (x - 3)^{2} = 0; x - 3 = 0; x = 3$

$2) x^{2} - 6x - 11 = 0$

$D = (-6)^{2} - 4 cdot 1 cdot (-11) = 36 + 44 = 80$

$x_{1,2} = dfrac{6 pm 4sqrt{5}}{2} = dfrac{2(3 pm 2sqrt{5})}{2} = 3 pm 2sqrt{5} = left[begin{array}{ccc}x_{1} = 3 - 2sqrt{5}\x_{2} = 3 + 2sqrt{5}\end{array}right$

Ответ: $x = 3; x = 3 - 2sqrt{5}; x = 3 +2sqrt{5}$